MOTION OF A PARTICLE 295 Evidently when m and mf are negligible compared with M \pfl - WV ' which is of Kepler's third law. PROBLEMS. 1. The gravitational acceleration at the surface of the earth is about 980 cm./sec.2 Calculate the mass and the average density of the earth, taking 6.4 X 108 cm. for the mean radius, and supposing it to attract as if all its mass were concentrated at its -center. 2. The periods of revolution of the earth and of the moon are, roughly, 365i and 27f days. Find the mass of the moon in tons. Take 6.0 X 1027 gm. for the mass of the earth. 3. The periods of revolution of the earth and of the moon are 365 J and 27J days, respectively, and the semi-major axes of their orbits are, approximately, 9.5 X 107 and 2.4 X 105 miles. Find the ratio of the mass of the sun to that of the earth. 4. Taking the period of the moon to be 27J days, and the radius of its orbit to be 3.85 X 1010 cm., show that the acceleration of the moon, due to the attraction of the earth, is equal to what would be expected from the gravitational law. Assume the gravitational acceleration at the surface of the earth, that is, at a point 6.4 X 10s cm. away from the center, to be 980 — -sec.2 6. Show that if the earth were suddenly stopped in its orbit it would fall into the sun in about 62.5 days. 6. Show that if a body is projected from the earth with a velocity of 7 miles per second it may leave the solar system. GENERAL PROBLEMS. 1. Find the expression for the central force under which a particle describes the orbit rn = an cos nd and consider the special cases when (a) n = i, (c) n = 1, (e) n = 2. 2. A particle moves in a central field of force with a velocity which is inversely proportional to the distance from the center of the field. Show that the orbit is a logarithmic spiral.